Buckling of multi-step non-uniform beams with elastically restrained boundary conditions

In this paper, the governing differential equation for buckling of a multi-step non-uniform beam under several concentrated axial forces is established. The special solutions of bending moment for buckling are found first for eight types of non-uniform beams, then the recurrence formula method and the transfer matrix method for determining the critical buckling force of a multi-step non-uniform beam are proposed. In the recurrence formula method, the two fundamental solutions of bending moment and the recurrence formulas developed in this paper are used to determine the critical buckling forces for a multi-step non-uniform beam without spring supports, and the four fundamental solutions of displacement and the recurrence formulas are adopted for buckling analysis of such a beam with rotational and translational springs at the ends and intermediate points. With this proposed procedure, the eigenvalue equation for a multi-step non-uniform beam with any kinds of the two end supports and any number of rotational and translational springs at the ends and intermediate points can be conveniently determined from a second order determinant. By using the transfer matrix method, the eigenvalue equation for the title problem can be established from a fourth order determinant. The numerical examples show: (1) the results calculated by the present methods are in good agreement with those obtained by FEM, (2) one of the advantages of the present methods is that the total number of the segments required by the proposed methods could be much less than that normally used in finite element method, (3) the two procedures proposed in this paper are exact and efficient methods. (C) 2001 Elsevier Science Ltd. All rights reserved.